How to Make It
In a bowl, whisk the chickpea flour and water until smooth. Whisk in the 1 tablespoon of olive oil and the salt. Cover and let stand at room temperature for at least 1 hour.
Heat 1/2 teaspoon of olive oil in an 8-inch nonstick skillet. Pour in about 1 1/2 tablespoons of the chickpea batter, tilting the skillet to distribute the batter evenly. Cook over moderate heat until browned and crisp on the bottom, about 2 minutes. Turn the cracker and brown the bottom, about 20 seconds. Transfer the cracker to a plate. Repeat with the remaining batter to make 5 more crackers.
Peel and devein 18 of the shrimp; reserve the shells. Cut the remaining unpeeled shrimp into 1/2-inch pieces. In a medium saucepan, heat the vegetable oil until shimmering. Add the shrimp pieces and the reserved shrimp shells and cook over high heat, stirring, until starting to brown, about 1 minute. Add the shallot and cook, stirring, until fragrant, about 1 minute. Add the tomato paste and cook, stirring, until shiny, about 1 minute. Add the cognac and boil for 1 minute. Add 2 cups of water and bring to a boil. Reduce the heat to moderately low and simmer for 12 minutes.
Transfer the contents of the saucepan to a food processor and process until the shells are finely ground. Pass the jus through a fine strainer into the saucepan. Boil the jus over high heat until reduced to 1/2 cup, about 7 minutes.
In a blender, puree the chickpeas with 3 tablespoons of water until smooth. Whisk the puree into the reduced shrimp jus and season with salt. Cook over moderate heat, stirring, until heated through.
In a large skillet, heat the olive oil. Add the shelled shrimp and cook over moderate heat, turning a few times, until just white throughout, about 3 minutes. Season lightly with salt and generously with pepper.
To serve, ladle the shrimp hummus into shallow bowls. Place the shrimp in the bowls, drizzling the peppery oil from the skillet around the shrimp. Arrange a chickpea cracker alongside the shrimp and serve.